You say you want to sort by value, but you don't have that in your code. Pass a lambda (or method reference) to sorted to tell it how you want to sort.

And you want to get the keys; use map to transform entries to keys.

List<Type> types = countByType.entrySet().stream()
        .sorted(Comparator.comparing(Map.Entry::getValue))
        .map(Map.Entry::getKey)
        .collect(Collectors.toList());

Based on Collectors documentation it's as simple as:

Map<String, Choice> result =
    choices.stream().collect(Collectors.toMap(Choice::getName,
                                              Function.identity()));

It looks like all you need is flatMap the values of the original Map and then continue processing as you currently do:

Collection< PublicationSession > wantedPublications = 
    effectiveDatePublicationMap.values() // Collection<List<PublicationSession>>
                               .stream() // Stream<List<PublicationSession>>
                               .flatMap(list->list.stream()) // Stream<PublicationSession>
                               .filter(pub -> PublicationStatus.valueOf(pub.getPublishStatus()) == PublicationStatus.COMPLETE)
                               .sorted(Comparator.comparing(PublicationSession::getCreateTime))
                               .collect(toMap(p -> p.getPublicationSession().getEffDateTime(), UnaryOperator.identity(), PICK_LATEST))
                               .values();